Ebers-Moll Model
The Ebers-Moll model is the classic large-signal description of a bipolar junction transistor, in which the device is represented by two back-to-back diodes (one for each junction) coupled by two current sources that carry the transport of carriers from one junction to the other. It was published in December 1954 in the Proceedings of the IRE by Jewell J. Ebers and John L. Moll, then both at Bell Telephone Laboratories, under the title "Large-signal behavior of junction transistors". Unlike small-signal models, it covers the whole operating range — cut-off, the active region, saturation and the reverse-active region — with one set of equations.
The model is built from the observation that a transistor is two p-n junctions sharing a thin base, each of which can be forward or reverse biased. The emitter-base diode carries current IES × (exp(VBE / VT) − 1) and the collector-base diode carries ICS × (exp(VBC / VT) − 1), where VT is the thermal voltage of about 26 millivolts and IES and ICS are the saturation currents of the two diodes. In normal forward-active operation the emitter junction is forward biased, and a fraction αF of its current — the forward alpha, about 0.98 to 0.998 — appears as collector current through a dependent source. If the roles of the two junctions are reversed, the collector junction injects and the emitter collects, with a reverse alpha αR, typically much smaller (from a few hundredths to about 0.5) because the collector is lightly doped and large in area. The model's equations are IC = αF × IF − IR and IE = −IF + αR × IR, with IF and IR being the two diode currents, and reciprocity requires that αF × IES = αR × ICS, so only three independent parameters are needed.
What the model explains is why a transistor can be both an amplifier and a switch within one description. In the active region the collector-diode term is negligible, giving IC ≈ αF × IES × exp(VBE / VT), which reveals the exponential control of collector current by base-emitter voltage and leads straight to the transconductance gm = IC / VT used in the hybrid-pi model. In saturation both exponentials are significant, and the collector current is reduced by the reverse-injected component, which accounts for the low collector-emitter saturation voltage of a few tenths of a volt and for the stored charge that delays turn-off. The same equations also allow the transistor to work "backward" with emitter and collector swapped, a mode with a weak alpha that is exploited in some logic circuits.
The model's limits led to the Gummel-Poon successor of 1970. Ebers-Moll is a first-order model that assumes constant alphas and ignores the dependence of base width on collector voltage (the Early effect), the fall of beta at low and high currents, and the series resistances of the base, emitter and collector. In 1970 H. K. Gummel and H. C. Poon, also of Bell Labs, published an integral charge-control model in the Bell System Technical Journal that expresses the collector current in terms of the total base charge, and so captures those effects with the same junction exponentials. The Gummel-Poon model, with extra parameters for the Early voltage, high-injection knee current and capacitances, became the default bipolar model in SPICE, the circuit simulator developed at the University of California, Berkeley in the early 1970s, and when its extra parameters are left at defaults it reduces to a version of Ebers-Moll. Later models such as VBIC and HICUM were developed for modern high-speed processes.
Practically, the model is used in two ways: as hand analysis and as the foundation of simulation. By hand, the exponential law gives the rule that base-emitter voltage changes about 60 millivolts per decade of collector current at room temperature and falls about 2 millivolts per degree Celsius — the origin of thermal bias drift. In simulation, it is the base from which more elaborate compact models are built. A common misconception is that alpha and beta are fixed numbers; in the model they are, but real devices vary, which is why the extended models were required. Another is that the model describes physical separate diodes: the two "diodes" share a single thin base, and a pair of discrete diodes wired back to back will not amplify, since the base must be thin enough that the carriers injected by one junction reach the other.
Read the Ebers-Moll model as the transistor's large-signal ledger: two diodes record what each junction injects, two current sources record what the base lets through to the other side, and every simpler model — hybrid-pi, h-parameters, the switch — is a region of that one description.